The sci-fi idea of a “warp drive” is based on simple physics: nothing can move faster than the speed of light. (There may be exceptions, such as in certain kinds of quantum tunneling, but this seems unlikely.) So, if we want a vessel to travel faster than light, we need to create a bubble that warps the space around a vessel without actually affecting that vessel.
A theory of warp travel put forward by in 1994 by Mexican physicist Miguel Alcubierre would involve a football-shaped ship with a giant ring around it, perhaps made from some kind of exotic metal. Space-time would be warped around the ship: less in front, more behind, creating an envelope of warped space while leaving the ship in “normal” space. Space-time itself would then “push” the vessel forward.
As this article points out, Alcubierre’s model had issues, but some new ideas are being kicked around that may provide some answers:
With this concept, [Alcubierre's] spacecraft would be able to achieve an effective speed of about 10 times the speed of light, all without breaking the cosmic speed limit.
The only problem is, previous studies estimated the warp drive would require a minimum amount of energy about equal to the mass-energy of the planet Jupiter.
But recently White calculated what would happen if the shape of the ring encircling the spacecraft was adjusted into more of a rounded donut, as opposed to a flat ring. He found in that case, the warp drive could be powered by a mass about the size of a spacecraft like the Voyager 1 probe NASA launched in 1977.
Furthermore, if the intensity of the space warps can be oscillated over time, the energy required is reduced even more, White found.
“The findings I presented today change it from impractical to plausible and worth further investigation,” White told SPACE.com. “The additional energy reduction realized by oscillating the bubble intensity is an interesting conjecture that we will enjoy looking at in the lab.”
As Marc Millis points out here, space-time itself may be able to move faster than light, as it is may have done in the Big Bang, and that’s what would power a warp drive. Honestly, I think we’re more likely to power faster-than-light travel on the sheer improbability of a warp drive, but that doesn’t mean we stop trying. It’s fascinating stuff that goes right to the origins of the universe.